We study the problem of optimal insurance contract design for risk management under a budget constraint. The contract holder takes into consideration that the loss distribution is not entirely known and therefore faces an ambiguity problem. For a given set of models, we formulate a minimax optimization problem of finding an optimal insurance contract that minimizes the distortion risk functional of the retained loss with premium limitation. We demonstrate that under the average value-at-risk measure, the entrance-excess of loss contracts are optimal under ambiguity, and we solve the distributionally robust optimal contract-design problem. It is assumed that the insurance premium is calculated according to a given baseline loss distribution and that the ambiguity set of possible distributions forms a neighborhood of the baseline distribution. To this end, we introduce a contorted Wasserstein distance. This distance is finer in the tails of the distributions compared to the usual Wasserstein distance.
Using two methods (ferric reducing antioxidant power and radical scavenging activity), the total polyphenol content (Folin–Ciocalteu reagent) and polyphenol patterns (HPLC) in 10 commercial lager beer brands produced in Romania was determined. Samples bottled in glass, plastic and aluminium packages were analysed for each brand when available. Results have indicated considerable variations in the total and individual phenolic contents as well as antioxidant activity across beer brands. Gallic and feulic acids were the dominant phenolic compounds identified samples. The statistical analysis indicates a significant difference between the levels of antioxidant activity detected using DPPH and FRAP, for all three types of material considered. The statistical test ANOVA indicates that the polyphenol concentration is similar across all types of material.
We examine a problem of demand for insurance indemnification, when the insured is sensitive to ambiguity and behaves according to the Maxmin-Expected Utility model of Gilboa and Schmeidler [43], whereas the insurer is a (risk-averse or risk-neutral) Expected-Utility maximizer. We characterize optimal indemnity functions both with and without the customary ex ante no-sabotage requirement on feasible indemnities, and for both concave and linear utility functions for the two agents. This allows us to provide a unifying framework in which we examine the effects of the no-sabotage condition, marginal utility of wealth, belief heterogeneity, as well as ambiguity (multiplicity of priors) on the structure of optimal indemnity functions. In particular, we show how the singularity in beliefs leads to an optimal indemnity function that involves full insurance on an event to which the insurer assigns zero probability, while the decision maker assigns a positive probability. We examine several illustrative examples, and we provide numerical studies for the case of a Wasserstein and a Rényi ambiguity set.
In this work, we provide robust bounds on the tail probabilities and the tail index of heavy-tailed distributions in the context of model misspecification. They are defined as the optimal value when computing the worst-case tail behavior over all models within some neighborhood of the reference model. The choice of the discrepancy between the models used to build this neighborhood plays a crucial role in assessing the size of the asymptotic bounds. We evaluate the robust tail behavior in ambiguity sets based on the Wasserstein distance and Csiszár f -divergence and obtain explicit expressions for the corresponding asymptotic bounds. In an application to Danish fire insurance claims we compare the difference between these bounds and show the importance of the choice of discrepancy measure.
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